// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

namespace Eigen {

namespace internal {
    template <typename _MatrixType> struct traits<ColPivHouseholderQR<_MatrixType>> : traits<_MatrixType>
    {
        typedef MatrixXpr XprKind;
        typedef SolverStorage StorageKind;
        typedef int StorageIndex;
        enum
        {
            Flags = 0
        };
    };

}  // end namespace internal

/** \ingroup QR_Module
  *
  * \class ColPivHouseholderQR
  *
  * \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the QR decomposition
  *
  * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
  * such that
  * \f[
  *  \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
  * \f]
  * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
  * upper triangular matrix.
  *
  * This decomposition performs column pivoting in order to be rank-revealing and improve
  * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
  *
  * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
  * 
  * \sa MatrixBase::colPivHouseholderQr()
  */
template <typename _MatrixType> class ColPivHouseholderQR : public SolverBase<ColPivHouseholderQR<_MatrixType>>
{
public:
    typedef _MatrixType MatrixType;
    typedef SolverBase<ColPivHouseholderQR> Base;
    friend class SolverBase<ColPivHouseholderQR>;

    EIGEN_GENERIC_PUBLIC_INTERFACE(ColPivHouseholderQR)
    enum
    {
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
    typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
    typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
    typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
    typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
    typedef HouseholderSequence<MatrixType, typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
    typedef typename MatrixType::PlainObject PlainObject;

private:
    typedef typename PermutationType::StorageIndex PermIndexType;

public:
    /**
    * \brief Default Constructor.
    *
    * The default constructor is useful in cases in which the user intends to
    * perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
    */
    ColPivHouseholderQR()
        : m_qr(), m_hCoeffs(), m_colsPermutation(), m_colsTranspositions(), m_temp(), m_colNormsUpdated(), m_colNormsDirect(), m_isInitialized(false),
          m_usePrescribedThreshold(false)
    {
    }

    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem \a size.
      * \sa ColPivHouseholderQR()
      */
    ColPivHouseholderQR(Index rows, Index cols)
        : m_qr(rows, cols), m_hCoeffs((std::min)(rows, cols)), m_colsPermutation(PermIndexType(cols)), m_colsTranspositions(cols), m_temp(cols),
          m_colNormsUpdated(cols), m_colNormsDirect(cols), m_isInitialized(false), m_usePrescribedThreshold(false)
    {
    }

    /** \brief Constructs a QR factorization from a given matrix
      *
      * This constructor computes the QR factorization of the matrix \a matrix by calling
      * the method compute(). It is a short cut for:
      *
      * \code
      * ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
      * qr.compute(matrix);
      * \endcode
      *
      * \sa compute()
      */
    template <typename InputType>
    explicit ColPivHouseholderQR(const EigenBase<InputType>& matrix)
        : m_qr(matrix.rows(), matrix.cols()), m_hCoeffs((std::min)(matrix.rows(), matrix.cols())), m_colsPermutation(PermIndexType(matrix.cols())),
          m_colsTranspositions(matrix.cols()), m_temp(matrix.cols()), m_colNormsUpdated(matrix.cols()), m_colNormsDirect(matrix.cols()), m_isInitialized(false),
          m_usePrescribedThreshold(false)
    {
        compute(matrix.derived());
    }

    /** \brief Constructs a QR factorization from a given matrix
      *
      * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
      *
      * \sa ColPivHouseholderQR(const EigenBase&)
      */
    template <typename InputType>
    explicit ColPivHouseholderQR(EigenBase<InputType>& matrix)
        : m_qr(matrix.derived()), m_hCoeffs((std::min)(matrix.rows(), matrix.cols())), m_colsPermutation(PermIndexType(matrix.cols())),
          m_colsTranspositions(matrix.cols()), m_temp(matrix.cols()), m_colNormsUpdated(matrix.cols()), m_colNormsDirect(matrix.cols()), m_isInitialized(false),
          m_usePrescribedThreshold(false)
    {
        computeInPlace();
    }

#ifdef EIGEN_PARSED_BY_DOXYGEN
    /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
      * *this is the QR decomposition, if any exists.
      *
      * \param b the right-hand-side of the equation to solve.
      *
      * \returns a solution.
      *
      * \note_about_checking_solutions
      *
      * \note_about_arbitrary_choice_of_solution
      *
      * Example: \include ColPivHouseholderQR_solve.cpp
      * Output: \verbinclude ColPivHouseholderQR_solve.out
      */
    template <typename Rhs> inline const Solve<ColPivHouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

    HouseholderSequenceType householderQ() const;
    HouseholderSequenceType matrixQ() const { return householderQ(); }

    /** \returns a reference to the matrix where the Householder QR decomposition is stored
      */
    const MatrixType& matrixQR() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return m_qr;
    }

    /** \returns a reference to the matrix where the result Householder QR is stored
     * \warning The strict lower part of this matrix contains internal values.
     * Only the upper triangular part should be referenced. To get it, use
     * \code matrixR().template triangularView<Upper>() \endcode
     * For rank-deficient matrices, use
     * \code
     * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
     * \endcode
     */
    const MatrixType& matrixR() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return m_qr;
    }

    template <typename InputType> ColPivHouseholderQR& compute(const EigenBase<InputType>& matrix);

    /** \returns a const reference to the column permutation matrix */
    const PermutationType& colsPermutation() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return m_colsPermutation;
    }

    /** \returns the absolute value of the determinant of the matrix of which
      * *this is the QR decomposition. It has only linear complexity
      * (that is, O(n) where n is the dimension of the square matrix)
      * as the QR decomposition has already been computed.
      *
      * \note This is only for square matrices.
      *
      * \warning a determinant can be very big or small, so for matrices
      * of large enough dimension, there is a risk of overflow/underflow.
      * One way to work around that is to use logAbsDeterminant() instead.
      *
      * \sa logAbsDeterminant(), MatrixBase::determinant()
      */
    typename MatrixType::RealScalar absDeterminant() const;

    /** \returns the natural log of the absolute value of the determinant of the matrix of which
      * *this is the QR decomposition. It has only linear complexity
      * (that is, O(n) where n is the dimension of the square matrix)
      * as the QR decomposition has already been computed.
      *
      * \note This is only for square matrices.
      *
      * \note This method is useful to work around the risk of overflow/underflow that's inherent
      * to determinant computation.
      *
      * \sa absDeterminant(), MatrixBase::determinant()
      */
    typename MatrixType::RealScalar logAbsDeterminant() const;

    /** \returns the rank of the matrix of which *this is the QR decomposition.
      *
      * \note This method has to determine which pivots should be considered nonzero.
      *       For that, it uses the threshold value that you can control by calling
      *       setThreshold(const RealScalar&).
      */
    inline Index rank() const
    {
        using std::abs;
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
        Index result = 0;
        for (Index i = 0; i < m_nonzero_pivots; ++i) result += (abs(m_qr.coeff(i, i)) > premultiplied_threshold);
        return result;
    }

    /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
      *
      * \note This method has to determine which pivots should be considered nonzero.
      *       For that, it uses the threshold value that you can control by calling
      *       setThreshold(const RealScalar&).
      */
    inline Index dimensionOfKernel() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return cols() - rank();
    }

    /** \returns true if the matrix of which *this is the QR decomposition represents an injective
      *          linear map, i.e. has trivial kernel; false otherwise.
      *
      * \note This method has to determine which pivots should be considered nonzero.
      *       For that, it uses the threshold value that you can control by calling
      *       setThreshold(const RealScalar&).
      */
    inline bool isInjective() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return rank() == cols();
    }

    /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
      *          linear map; false otherwise.
      *
      * \note This method has to determine which pivots should be considered nonzero.
      *       For that, it uses the threshold value that you can control by calling
      *       setThreshold(const RealScalar&).
      */
    inline bool isSurjective() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return rank() == rows();
    }

    /** \returns true if the matrix of which *this is the QR decomposition is invertible.
      *
      * \note This method has to determine which pivots should be considered nonzero.
      *       For that, it uses the threshold value that you can control by calling
      *       setThreshold(const RealScalar&).
      */
    inline bool isInvertible() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return isInjective() && isSurjective();
    }

    /** \returns the inverse of the matrix of which *this is the QR decomposition.
      *
      * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
      *       Use isInvertible() to first determine whether this matrix is invertible.
      */
    inline const Inverse<ColPivHouseholderQR> inverse() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return Inverse<ColPivHouseholderQR>(*this);
    }

    inline Index rows() const { return m_qr.rows(); }
    inline Index cols() const { return m_qr.cols(); }

    /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
      *
      * For advanced uses only.
      */
    const HCoeffsType& hCoeffs() const { return m_hCoeffs; }

    /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
      * who need to determine when pivots are to be considered nonzero. This is not used for the
      * QR decomposition itself.
      *
      * When it needs to get the threshold value, Eigen calls threshold(). By default, this
      * uses a formula to automatically determine a reasonable threshold.
      * Once you have called the present method setThreshold(const RealScalar&),
      * your value is used instead.
      *
      * \param threshold The new value to use as the threshold.
      *
      * A pivot will be considered nonzero if its absolute value is strictly greater than
      *  \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
      * where maxpivot is the biggest pivot.
      *
      * If you want to come back to the default behavior, call setThreshold(Default_t)
      */
    ColPivHouseholderQR& setThreshold(const RealScalar& threshold)
    {
        m_usePrescribedThreshold = true;
        m_prescribedThreshold = threshold;
        return *this;
    }

    /** Allows to come back to the default behavior, letting Eigen use its default formula for
      * determining the threshold.
      *
      * You should pass the special object Eigen::Default as parameter here.
      * \code qr.setThreshold(Eigen::Default); \endcode
      *
      * See the documentation of setThreshold(const RealScalar&).
      */
    ColPivHouseholderQR& setThreshold(Default_t)
    {
        m_usePrescribedThreshold = false;
        return *this;
    }

    /** Returns the threshold that will be used by certain methods such as rank().
      *
      * See the documentation of setThreshold(const RealScalar&).
      */
    RealScalar threshold() const
    {
        eigen_assert(m_isInitialized || m_usePrescribedThreshold);
        return m_usePrescribedThreshold ? m_prescribedThreshold
                                          // this formula comes from experimenting (see "LU precision tuning" thread on the list)
                                          // and turns out to be identical to Higham's formula used already in LDLt.
                                          :
                                          NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
    }

    /** \returns the number of nonzero pivots in the QR decomposition.
      * Here nonzero is meant in the exact sense, not in a fuzzy sense.
      * So that notion isn't really intrinsically interesting, but it is
      * still useful when implementing algorithms.
      *
      * \sa rank()
      */
    inline Index nonzeroPivots() const
    {
        eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
        return m_nonzero_pivots;
    }

    /** \returns the absolute value of the biggest pivot, i.e. the biggest
      *          diagonal coefficient of R.
      */
    RealScalar maxPivot() const { return m_maxpivot; }

    /** \brief Reports whether the QR factorization was successful.
      *
      * \note This function always returns \c Success. It is provided for compatibility
      * with other factorization routines.
      * \returns \c Success
      */
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "Decomposition is not initialized.");
        return Success;
    }

#ifndef EIGEN_PARSED_BY_DOXYGEN
    template <typename RhsType, typename DstType> void _solve_impl(const RhsType& rhs, DstType& dst) const;

    template <bool Conjugate, typename RhsType, typename DstType> void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif

protected:
    friend class CompleteOrthogonalDecomposition<MatrixType>;

    static void check_template_parameters() { EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar); }

    void computeInPlace();

    MatrixType m_qr;
    HCoeffsType m_hCoeffs;
    PermutationType m_colsPermutation;
    IntRowVectorType m_colsTranspositions;
    RowVectorType m_temp;
    RealRowVectorType m_colNormsUpdated;
    RealRowVectorType m_colNormsDirect;
    bool m_isInitialized, m_usePrescribedThreshold;
    RealScalar m_prescribedThreshold, m_maxpivot;
    Index m_nonzero_pivots;
    Index m_det_pq;
};

template <typename MatrixType> typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
{
    using std::abs;
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
    return abs(m_qr.diagonal().prod());
}

template <typename MatrixType> typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
{
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
    return m_qr.diagonal().cwiseAbs().array().log().sum();
}

/** Performs the QR factorization of the given matrix \a matrix. The result of
  * the factorization is stored into \c *this, and a reference to \c *this
  * is returned.
  *
  * \sa class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
  */
template <typename MatrixType>
template <typename InputType>
ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const EigenBase<InputType>& matrix)
{
    m_qr = matrix.derived();
    computeInPlace();
    return *this;
}

template <typename MatrixType> void ColPivHouseholderQR<MatrixType>::computeInPlace()
{
    check_template_parameters();

    // the column permutation is stored as int indices, so just to be sure:
    eigen_assert(m_qr.cols() <= NumTraits<int>::highest());

    using std::abs;

    Index rows = m_qr.rows();
    Index cols = m_qr.cols();
    Index size = m_qr.diagonalSize();

    m_hCoeffs.resize(size);

    m_temp.resize(cols);

    m_colsTranspositions.resize(m_qr.cols());
    Index number_of_transpositions = 0;

    m_colNormsUpdated.resize(cols);
    m_colNormsDirect.resize(cols);
    for (Index k = 0; k < cols; ++k)
    {
        // colNormsDirect(k) caches the most recent directly computed norm of
        // column k.
        m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
        m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
    }

    RealScalar threshold_helper = numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
    RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());

    m_nonzero_pivots = size;  // the generic case is that in which all pivots are nonzero (invertible case)
    m_maxpivot = RealScalar(0);

    for (Index k = 0; k < size; ++k)
    {
        // first, we look up in our table m_colNormsUpdated which column has the biggest norm
        Index biggest_col_index;
        RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols - k).maxCoeff(&biggest_col_index));
        biggest_col_index += k;

        // Track the number of meaningful pivots but do not stop the decomposition to make
        // sure that the initial matrix is properly reproduced. See bug 941.
        if (m_nonzero_pivots == size && biggest_col_sq_norm < threshold_helper * RealScalar(rows - k))
            m_nonzero_pivots = k;

        // apply the transposition to the columns
        m_colsTranspositions.coeffRef(k) = biggest_col_index;
        if (k != biggest_col_index)
        {
            m_qr.col(k).swap(m_qr.col(biggest_col_index));
            std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
            std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
            ++number_of_transpositions;
        }

        // generate the householder vector, store it below the diagonal
        RealScalar beta;
        m_qr.col(k).tail(rows - k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);

        // apply the householder transformation to the diagonal coefficient
        m_qr.coeffRef(k, k) = beta;

        // remember the maximum absolute value of diagonal coefficients
        if (abs(beta) > m_maxpivot)
            m_maxpivot = abs(beta);

        // apply the householder transformation
        m_qr.bottomRightCorner(rows - k, cols - k - 1)
            .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows - k - 1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k + 1));

        // update our table of norms of the columns
        for (Index j = k + 1; j < cols; ++j)
        {
            // The following implements the stable norm downgrade step discussed in
            // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
            // and used in LAPACK routines xGEQPF and xGEQP3.
            // See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
            if (m_colNormsUpdated.coeffRef(j) != RealScalar(0))
            {
                RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
                temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
                temp = temp < RealScalar(0) ? RealScalar(0) : temp;
                RealScalar temp2 = temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) / m_colNormsDirect.coeffRef(j));
                if (temp2 <= norm_downdate_threshold)
                {
                    // The updated norm has become too inaccurate so re-compute the column
                    // norm directly.
                    m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
                    m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
                }
                else
                {
                    m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
                }
            }
        }
    }

    m_colsPermutation.setIdentity(PermIndexType(cols));
    for (PermIndexType k = 0; k < size /*m_nonzero_pivots*/; ++k)
        m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));

    m_det_pq = (number_of_transpositions % 2) ? -1 : 1;
    m_isInitialized = true;
}

#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename _MatrixType>
template <typename RhsType, typename DstType>
void ColPivHouseholderQR<_MatrixType>::_solve_impl(const RhsType& rhs, DstType& dst) const
{
    const Index nonzero_pivots = nonzeroPivots();

    if (nonzero_pivots == 0)
    {
        dst.setZero();
        return;
    }

    typename RhsType::PlainObject c(rhs);

    c.applyOnTheLeft(householderQ().setLength(nonzero_pivots).adjoint());

    m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots).template triangularView<Upper>().solveInPlace(c.topRows(nonzero_pivots));

    for (Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
    for (Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
}

template <typename _MatrixType>
template <bool Conjugate, typename RhsType, typename DstType>
void ColPivHouseholderQR<_MatrixType>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const
{
    const Index nonzero_pivots = nonzeroPivots();

    if (nonzero_pivots == 0)
    {
        dst.setZero();
        return;
    }

    typename RhsType::PlainObject c(m_colsPermutation.transpose() * rhs);

    m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
        .template triangularView<Upper>()
        .transpose()
        .template conjugateIf<Conjugate>()
        .solveInPlace(c.topRows(nonzero_pivots));

    dst.topRows(nonzero_pivots) = c.topRows(nonzero_pivots);
    dst.bottomRows(rows() - nonzero_pivots).setZero();

    dst.applyOnTheLeft(householderQ().setLength(nonzero_pivots).template conjugateIf<!Conjugate>());
}
#endif

namespace internal {

    template <typename DstXprType, typename MatrixType>
    struct Assignment<DstXprType,
                      Inverse<ColPivHouseholderQR<MatrixType>>,
                      internal::assign_op<typename DstXprType::Scalar, typename ColPivHouseholderQR<MatrixType>::Scalar>,
                      Dense2Dense>
    {
        typedef ColPivHouseholderQR<MatrixType> QrType;
        typedef Inverse<QrType> SrcXprType;
        static void run(DstXprType& dst, const SrcXprType& src, const internal::assign_op<typename DstXprType::Scalar, typename QrType::Scalar>&)
        {
            dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
        }
    };

}  // end namespace internal

/** \returns the matrix Q as a sequence of householder transformations.
  * You can extract the meaningful part only by using:
  * \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
template <typename MatrixType> typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>::householderQ() const
{
    eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
    return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
}

/** \return the column-pivoting Householder QR decomposition of \c *this.
  *
  * \sa class ColPivHouseholderQR
  */
template <typename Derived> const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::colPivHouseholderQr() const
{
    return ColPivHouseholderQR<PlainObject>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
